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The aim of this paper is to provide new stability results for sequences of metric measure spaces (Xᵢ, dᵢ, mᵢ) convergent in the measured Gromov-Hausdorff sense. By adopting the so-called extrinsic approach of embedding all metric spaces into a common one (X, d), we extend the results of Gigli-Mondino-Savar\'e by providing Mosco convergence of Cheeger's energies and compactness theorems in the whole range of Sobolev spaces H^1, p, including the space BV, and even with a variable exponent pᵢ 1,. In addition, building on the results of Ambrosio-Stra-Trevisan, we provide local convergence results for gradient derivations. We use these tools to improve the spectral stability results, previously known for p>1 and for Ricci limit spaces, getting continuity of Cheeger's constant. In the dimensional case N<, we improve some rigidity and almost rigidity results by Ketterer and Cavaletti-Mondino. On the basis of the second-order calculus by Gigli, in the class of RCD (K, ) spaces we provide stability results for Hessians and W^2, 2 functions and we treat the stability of the Bakry-\'Emery condition BE (K, N) and of Ric KI, with K and N not necessarily constant.
Ambrosio et al. (Tue,) studied this question.